1. Introduction

Without going into details, I have built a model that identifies “latent communities” (LC) in the meadows data. There is no intention to classify sites in the NVC sense. Instead, the idea is that the latent communities may be more or less expressed at any site. So what follows is a description of the latent communities, and an analysis of their expression by survey site.

At a quick look, it appears to me that the LC expressions make at least some sense in terms of what we already know about the sites. If that impression holds up to further scrutiny, then I think that the model will help us to relate the patterns that we see to physical, ecological and historical aspects of the environment. I have tried to encapsulate this idea in Figure 1. Here we have the latent communities on the left and their manifestation at the survey sites on the right. In between, the boxes labelled “Environment” are what is called in the literature a “filtering framework”. I quote: “the abiotic conditions define the environmental filters selecting species from a regional species pool …” (Munkemiller et al 2020). My scheme agrees with this in the presence of a filtering framework, but the latent communities are NOT the same as a species pool, also known as a Community Pool (CP). Here’s the difference:

The conceptual distinction is significant:

Figure 1.

2. Latent communities

According to this scheme, a latent community is a set of relationships between pairs of plants such that they occur together in the data more or less frequently than would be expected by chance (p < 0.05). These plant pairs are called dyads, and they may be associative (occur together more frequently than would be expected) or dissociative.

Dyads, not plants, are counted in assessing latent community expression at a site; in order count towards LC expression, both elements of the dyad must be present, and the contribution that they make is determined by the sign (associative/dissociative) of the link between them, see Calculation of LC expression below.

Table 1. Model parameters
LC1 LC2 LC3 LC4 LC5 LC6 LC7 LC8
0.858 0.517 0.223 0.481 0.071 0.022 0.188 0.003
0.517 0.596 0.169 0.110 0.226 0.055 0.080 0.009
0.223 0.169 0.635 0.024 0.049 0.254 0.120 0.014
0.481 0.110 0.024 0.565 0.071 0.085 0.072 0.008
0.071 0.226 0.049 0.071 0.260 0.079 0.011 0.017
0.022 0.055 0.254 0.085 0.079 0.206 0.011 0.030
0.188 0.080 0.120 0.072 0.011 0.011 0.076 0.015
0.003 0.009 0.014 0.008 0.017 0.030 0.015 0.035

The dyads are the observed data. The statistical model (Stochastic Block Model) builds a representation of the uncertain nature of the data by assigning probabilities to the links. The probabilities form blocks such that the probability of links between dyads within a block differs from the probability of links between dyads ending in different blocks. Generally, in-block probabilities are greater than out_block probabilities, forming positive groups of dyads. In our case, there are eight blocks. The model is summed up by an 8x8 matrix. The values on the leading diagonal are the in-block probabilities, the off-diagonal values are the probabilities of finding links between the corresponding blocks. The matrix is shown in Table 1. The in_block probabilities in the leading diagonal are shown in bold face, and the text is red when the in_block probability is greater than the out_block probabilities. Latent communities (blocks) 1 - 5 are good, 6 needs to be treated with caution, and the probabilities for 7 and 8 are so low that they may best be ignored (the probabilities are about 1/3 that of LC6, and they contain species that are poorly represented in the data).

I don’t want to go into details of the Stochastic Block Model here, but it is worth mentioning that it is not told anything about the associative or dissociative nature of the links it is modelling. That information is added later.

Details of the Latent Communities follow. For each, there is a graph and a table. The graphs are useful in visualising the contribution of associative and dissociative links. The size of the species symbols is an indication of the species frequency in the data. The tables list the species in the dyads of which the LC is composed.

Latent Community 1.

Latent Community 1 species
name count frequency
Anemone_nemorosa 11 6.96
Carex_caryophyllea 12 7.59
Carex_flacca 35 22.2
Lathyrus_montanus 9 5.70
Leontodon_hispidus 13 8.23
Lolium_perenne 111 70.3
Pimpinella_saxifraga 17 10.8
Potentilla_erecta 27 17.1
Potentilla_sterilis 9 5.70
Pseudoscleropodium_purum 23 14.6
Rhinanthus_minor 11 6.96
Rhytidiadelphus_squarrosus 36 22.8
Stachys_betonica 12 7.59
Succisa_pratensis 19 12.0

Latent Community 2.

Latent Community 2 species
name count frequency
Achillea_millefolium 71 44.9
Ajuga_reptans 42 26.6
Alopecurus_pratensis 78 49.4
Calliergon_cuspidatum 50 31.6
Centaurea_nigra 71 44.9
Conopodium_majus 21 13.3
Dactylorhiza_fuchsii 28 17.7
Genista_tinctoria 9 5.70
Hypochaeris_radicata 50 31.6
Leucanthemum_vulgare 35 22.2
Lotus_corniculatus 129 81.6
Lotus_uliginosus 102 64.6
Luzula_campestris 79 50
Plantago_lanceolata 59 37.3
Prunella_vulgaris 88 55.7
Ranunculus_bulbosus 30 19.0
Veronica_chamaedrys 58 36.7

Latent Community 3.

Latent Community 3 species
name count frequency
Achillea_ptarmica 7 4.43
Angelica_sylvestris 13 8.23
Carex_panicea 6 3.80
Cirsium_palustre 46 29.1
Equisetum_arvense 13 8.23
Galium_palustre 16 10.1
Hypericum_tetrapterum 11 6.96
Juncus_acutiflorus 26 16.5
Juncus_articulatus 11 6.96
Juncus_conglomeratus 29 18.4
Juncus_effusus 43 27.2
Lychnis_flos-cuculi 7 4.43
Mentha_aquatica 6 3.80
Oenanthe_crocata 8 5.06
Pulicaria_dysenterica 20 12.7
Ranunculus_flammula 8 5.06

Latent Community 4.

Latent Community 4 species
name count frequency
Agrostis_canina_spp_montana 15 9.49
Briza_media 6 3.80
Cerastium_fontanum 134 84.8
Cirsium_arvense 94 59.5
Danthonia_decumbens 6 3.80
Equisetum_telmateia 3 1.90
Hieracium_pilosella 7 4.43
Hyacinthoides_non-scripta 12 7.59
Poa_trivialis 122 77.2
Ranunculus_acris 140 88.6
Ranunculus_repens 145 91.8
Trifolium_repens 140 88.6
Veronica_officinalis 6 3.80
Viola_riviniana 8 5.06

Latent Community 5.

Latent Community 5 species
name count frequency
Anthoxanthum_odoratum 144 91.1
Arrhenatherum_elatius 25 15.8
Bellis_perennis 8 5.06
Brachythecium_rutabulum 117 74.1
Bromus_hordeaceus 27 17.1
Carex_hirta 48 30.4
Cerastium_glomeratum 8 5.06
Cynosurus_cristatus 54 34.2
Dactylis_glomerata 105 66.5
Eurhynchium_praelongum 56 35.4
Festuca_rubra 106 67.1
Geranium_dissectum 31 19.6
Heracleum_sphondylium 26 16.5
Lathyrus_pratensis 78 49.4
Phleum_pratense 85 53.8
Plantago_major 8 5.06
Potentilla_reptans 59 37.3
Rumex_acetosa 130 82.3
Stellaria_graminea 120 75.9
Taraxacum_officinale 109 69.0
Tragopogon_pratensis 6 3.80
Trifolium_dubium 34 21.5
Trifolium_pratense 92 58.2
Veronica_serpyllifolia 57 36.1
Vicia_cracca 17 10.8
Vicia_sativa 17 10.8
Vicia_tetrasperma 19 12.0

Latent Community 6.

Latent Community 6 species
name count frequency
Agrimonia_eupatoria 16 10.1
Calystegia_sepium 4 2.53
Cardamine_flexuosa 3 1.90
Cardamine_pratensis 58 36.7
Carex_ovalis 21 13.3
Cruciata_laevipes 7 4.43
Dactylorhiza_praetermissa 3 1.90
Deschampsia_cespitosa 12 7.59
Festuca_arundinacea 11 6.96
Festuca_pratensis 24 15.2
Filipendula_ulmaria 12 7.59
Galium_aparine 17 10.8
Glechoma_hederacea 15 9.49
Glyceria_fluitans 7 4.43
Juncus_inflexus 9 5.70
Lapsana_communis 3 1.90
Myosotis_discolor 8 5.06
Poa_annua 7 4.43
Ranunculus_ficaria 11 6.96
Rubus_fruticosus 16 10.1
Rumex_crispus 15 9.49
Rumex_obtusifolius 36 22.8
Senecio_jacobaea 28 17.7
Stachys_sylvatica 3 1.90
Urtica_dioica 13 8.23
Veronica_beccabunga 4 2.53

Latent Community 7.

Latent Community 7 species
name count frequency
Agrostis_canina 22 13.9
Carex_pilulifera 3 1.90
Holcus_mollis 7 4.43
Hypericum_perforatum 4 2.53
Hypericum_pulchrum 5 3.16
Lathyrus_nissolia 4 2.53
Luzula_multiflora 8 5.06
Mentha_arvensis 5 3.16
Pedicularis_sylvatica 3 1.90
Primula_veris 5 3.16
Pteridium_aquilinum 5 3.16
Rumex_acetosella 3 1.90
Silaum_silaus 9 5.70
Stellaria_alsine 5 3.16

Latent Community 8.

Latent Community 8 species
name count frequency
Alopecurus_geniculatus 11 6.96
Atrichum_undulatum 4 2.53
Carex_laevigata 2 1.27
Carex_otrubae 2 1.27
Cirsium_vulgare 5 3.16
Convolvulus_arvensis 6 3.80
Crepis_capillaris 4 2.53
Dactylorhiza_maculata 2 1.27
Dicranella_staphylina 2 1.27
Elymus_repens 17 10.8
Epilobium_hirsutum 3 1.90
Epilobium_tetragonum 2 1.27
Equisetum_fluviatile 3 1.90
Equisetum_palustre 2 1.27
Euphrasia_nemorosa 2 1.27
Festuca_ovina 2 1.27
Fissidens_taxifolius 6 3.80
Galium_saxatile 2 1.27
Galium_verum 3 1.90
Geum_urbanum 2 1.27
Hordeum_secalinum 13 8.23
Juncus_acutiflorus_articulatus 6 3.80
Juncus_bufonius 3 1.90
Leontodon_autumnalis 2 1.27
Leontodon_taraxacoides 5 3.16
Lysimachia_nummularia 2 1.27
Malva_moschata 9 5.70
Molinia_caerulea 2 1.27
Myosotis_arvensis 4 2.53
Oenanthe_pimpinelloides 4 2.53
Ophioglossum_vulgatum 10 6.33
Poa_pratensis 17 10.8
Potentilla_anserina 3 1.90
Potentilla_hybrid 4 2.53
Ranunculus_sardous 2 1.27
Rumex_conglomeratus 6 3.80
Rumex_sanguineus 2 1.27
Senecio_aquaticus 2 1.27
Senecio_erucifolius 4 2.53
Sonchus_oleraceus 3 1.90
Stachys_palustris 2 1.27
Trifolium_medium 2 1.27
Trisetum_flavescens 2 1.27
Veronica_filiformis 5 3.16
Veronica_persica 2 1.27
Vicia_hirsuta 3 1.90
Vulpia_bromoides 2 1.27

3. Latent Community expression at survey sites.

Here is a summary of all the Latent Communities expressed at all the survey sites. Two features are immediately obvious:

  1. Several LC may be expressed at a single site (Wood Field, Bushy Field, Lindfield Bridge).
  2. Some sites have very low expression of any LC (Long Mead SE, SW; Pond field W). We can investigate these sites to find out what is going on.

The pages following show the expressions of individual communities by site. The scale on the left hand side is the percentage of the maximum possible expression.

Latent Community 1.

Latent Community 2.

Latent Community 3.

Latent Community 4.

Latent Community 5.

Latent Community 6.

Latent Community 7.

Latent Community 8.

4. Calculation of LC expression.

  1. Write Gl for the graph of Latent Community l, of degree rl (Latent communities are expressed as graphs in section 2 above. The degree of a graph is the number of links between the nodes; in this case the nodes are plant species.)
  2. Write Gls for the subgraph of Gl represented at site s.
  3. Write Als for the subgraph of Gls with associative links, and Dls for the subgraph of Gls with dissociative links.

Then the expression Xls of Latent Community l at site s is Xls = 100(als-dls)/rl

where als is the degree (number of links) of Als and dls is the degree of Dls.

The presence of associative dyads increases Xls while dissociative dyads decrease Xls. An LC with dissociative links can never be 100% represented; another option would be to normalise against the degree of Al, the number of associative links in the parent LC. I decided against this because it is conceivable that there could be an LC with only dissociative links. Negative values of Xls are possible.

5. Where next?

Can we glean any understanding of community organisation in our data by examining the species composition of the latent communities, and their site-specific expressions? There are hints of some possibilities:

  • LC1 could be characterised as dyads with any of several rather interesting plants, but NOT paired with Lolium perenne. The meadows where LC1 is most fully expressed are Gravetye East, Gravetye West and Plain Field North. I think I am correct that these are fields with a management history of intentional meadow restoration, and that would explain the presence of interesting plants; to the extent that L perenne is present at these sites (and I need to check up whether it is or not), it could be interpreted as a relic from former more intensive management.

  • Latent Community 3 contains an assemblage of plants of damp or wet habitats. It is particularly expressed at Bushy Field, Daltons Meadow, Hanging Meadow, The Mead and White Coppice. This makes sense as these are all damp but well managed (from a wildflower meadow perspective).

  • LC6 also includes plants from wet habitats, but maybe with a suggestion of higher nutrient level (Urtica dioica, Rumex crispus, Rumex acutifolius). Lindfield Bridge has the strongest expression of LC6; it is a riparine bank bordering a former agricultural field. It was pointed out in Section 2 that LC6 is anomalous in that the in_block probability is less than one of the out_block probabilities; in fact, less than the out_block probability with the other damp community, LC3, suggesting that these communities have a mutually exclusive relationship.

It will be interesting to use this tool to explore the data for further evidence of biotic or abiotic interactions which could further be tested (in principle at least) by field work or reference to literature.

6. Bibliography.

Munkemiller et al. 2020. Dos and Don’ts When Inferring Assembly Rules from Diversity Patterns. https://onlinelibrary.wiley.com/doi/full/10.1111/geb.13098.